In this paper, we give some results on the existence of $L_{loc}^p$-solutions of nonlinear impulsive Volterra integral equations in Banach spaces and on the structure of the set of $L_{loc}^p$-solutions by the theory of Fréchet spaces.

The exponential integral $\text{e}\text{i}(\text{λ}x)$ and its associated functions ${\text{ei}}_{+}(\text{λ}x)$ and ${\text{ei}}_{-}(\text{λ}x)$ are defined as locally summable functions on the real line and their derivatives are found as distributions. Some convolution products of these distributions and other distributions are then found.

Let ${ℒ}_A$ be the usual language for arithmetic. Let $\phi (x)$ be an ${ℒ}_A$-formula. $\phi (x)$ may contain free variables distinct from $x$ as parameters. We consider the following two schemata.

They are called the induction schema and the least number principle, respectively. IOpen, LOpen will denote the theory $P{A}^{-}\cup \left\{I_{\phi (x)}\right|\phi (x):open\}$, $P{A}^{-}{\cup }^{\text{}}\left\{L_{\phi (x)}\right|\phi (x):open\}$, respectively.

In this paper we prove the equivalence of IOpen and LOpen. Van den Dries [v.d.D] noted that this can be proven model theoretically by using ideas in the proof of Shepherdson’s theorem in [S1]. Our proof is syntactical and not model theoretical.

From a Minkowski-type metric on $R_{+}^n$ satisfying the Einstein condition, we derived a nonlinear partial differential equation. We tried to get some numerical approximate solution with certain boundary conditions by the finite element method in [10]. Regarding the exact solution, we shall give a fundamental theorem, which tells us the above numerical approximate solution is not worth to using the word “approximate”.

We prove that results for discrete weighted Hardy inequality can be obtained from a two-weight inequality for the classical Hardy operator.

Given two closed linear operators T and A in a Banach space, a sufficient condition is presented for the family , to be holomorphic of type (A). Detailed results are established when T and A are m-accretive in a reflexive Banach space. The results restricted to the Hilbert space case are almost identical with Kato’s. As an application a simple first-order singular differential operator in the $L^p$-space is discussed. This is a generalization of Kato’s result in the $L^2$-case.

In this paper, we investigate what kind of modal logics can be constructed into the Gentzen-type formulation which enjoys the Cut elimination theorem. We treat normal, regular and monotonic modal logics. A normal modal logic is the modal logic with the Rule of Necessitation and the axiom $\text{K}:\square (A\supset B)\supset (\square A\supset \square B)$, a regular modal logic with the rule of inference $\text{RR} :\frac{A,Г\to B}{\square A,\square Г\to \square B}$ and a monotonic modal logic with the rule of inference $\text{RM} :\frac{A\to B}{\square A\to \square B}$. We provide Gentzen-type formulation and prove the cut elimination theorem for the following modal logics: RT4 (a regular modal logic with the axioms $\text{T}:\square A\supset A$ and 4:$\square A\supset \square \square A)$, MT4 (a monotonic modal logic with the axioms T and 4), MNT4 (a monotonic modal logic with the Rule of Necessitation and the axioms T and 4), RT (a regular modal logic with the axiom T), MT (a monotonic modal logic with the axiom T), MNT (a monotonic modal logic with the Rule of Necessitation and the axiom T), KL (a normal modal logic with the axiom $\text{L}:\square A\supset (A\vee \square B))$ and RTA (a regular modal logic with axioms T and .

We define a functional constructed from the scalar curvature of a certain metric on a principal fiber bundle and obtain some equations which correspond to the Einstein field equation, the Yang-Mills equation and the Brans-Dicke type wave equation by variations of this functional.

Let $M$ be a Kählerian manifold and $\nabla$ the Levi-Civita connection of $M$. In this paper, we consider a linear connection ${\nabla }^0$ having a certain relation to $\nabla$ and a Kählerian torse-forming vector field on $M$. The properties of the curvature tensor $R^0$ of ${\nabla }^0$ and the Bochner curvature tensors are studied. Also we apply these properties to a Kählerian submersion.

This paper is concerned with two-symbol orthogonal arrays of strength 2, size 24, 6 constraints and index 6 which are obtained by selecting 6 columns from saturated two-symbol orthogonal arrays having maximal constraints. In [15], it has been shown that there are precisely 1,317 isomorphic classes of such arrays with respect to the permutation of factors and levels within factors. In this paper, the statistical properties of those orthogonal 24-run $2^6$-factorial designs which are associated with the above are investigated.

By the Schauder’s fixed point theorem the ${\mathfrak{L}}_p$-equivalence between two impulse differential equations is proved.